class: center, middle, inverse, title-slide .title[ # Three common mistakes in statistics and how to avoid them ] .author[ ### Elizabeth Pankratz ] .institute[ ### Department of Psychology<br>The University of Edinburgh ] --- ## Something you won't be able to unsee: -- .pull-left[  Reeder et al. (2017) in **Journal of Memory and Language.** ] .pull-right[  Elazar et al. (2022) in **Cognitive Science.** <br>  Harrigan et al. (2022) in **Language.** ] --- .pull-left[ ## The mistake ] .pull-right[ ## How you'll avoid it ] -- .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3L562.7 256l-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg> **A common R mistake:** Letting R treat all variables that consist of numbers as continuous numeric. ] .pull-right[ ] -- .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M160 80c0-26.5 21.5-48 48-48h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H208c-26.5 0-48-21.5-48-48V80zM0 272c0-26.5 21.5-48 48-48H80c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V272zM368 96h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H368c-26.5 0-48-21.5-48-48V144c0-26.5 21.5-48 48-48z"/></svg> **An advanced stats mistake:** Modelling categorical, ordinal data as if it were numeric. ] .pull-right[ ] -- .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M192 32c17.7 0 32 14.3 32 32V199.5l111.5-66.9c15.2-9.1 34.8-4.2 43.9 11s4.2 34.8-11 43.9L254.2 256l114.3 68.6c15.2 9.1 20.1 28.7 11 43.9s-28.7 20.1-43.9 11L224 312.5V448c0 17.7-14.3 32-32 32s-32-14.3-32-32V312.5L48.5 379.4c-15.2 9.1-34.8 4.2-43.9-11s-4.2-34.8 11-43.9L129.8 256 15.5 187.4c-15.2-9.1-20.1-28.7-11-43.9s28.7-20.1 43.9-11L160 199.5V64c0-17.7 14.3-32 32-32z"/></svg> **A foundational stats mistake:** Interpreting a significant *p*-value as evidence that an effect exists in the real world. ] .pull-right[ ] --- ### The data: Students' anxiety ratings for "Going to ask my statistics teacher for individual help with material I am having difficulty understanding." <img src="data:image/png;base64,#demo_files/figure-html/bar-aggregated-1.png" width="576" style="display: block; margin: auto;" /> ] --- ### The data: Students' anxiety ratings for "Going to ask my statistics teacher for individual help with material I am having difficulty understanding." .pull-left[ ``` r slice(anx, 45:50) ``` ``` ## # A tibble: 6 × 3 ## unique_id gender rating ## <chr> <chr> <dbl> ## 1 7d28c303 Female/Woman 4 ## 2 7d55383a Another Gender 4 ## 3 8116550a Female/Woman 1 ## 4 83491ff9 Female/Woman 4 ## 5 8450f8ad Male/Man 2 ## 6 876547d6 Female/Woman 3 ``` ] -- .pull-right[ `rating` looks like numbers, and R treats it like numbers. And we can manipulate it like numbers. ``` r mean(anx$rating) ``` ``` ## [1] 2.868054 ``` ] --- ## Why Likert scale ratings are not continuous numeric .center[  ] --- count:false ## Why Likert scale ratings are not continuous numeric .center[  ] --- count:false ## Why Likert scale ratings are not continuous numeric .center[  ] --- count:false ## Why Likert scale ratings are not continuous numeric .center[  ] --- ## Remember: We are smarter than R is Store categorical variables as factors. ``` r anx <- anx |> mutate(rating = factor(rating)) ``` -- Now it's impossible to incorrectly treat them as if they're numeric! ``` r mean(anx$rating) ``` ``` ## [1] NA ``` --- .pull-left[ ## The mistake ] .pull-right[ ## How you'll avoid it ] .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3L562.7 256l-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg> **A common R mistake:** Letting R treat all variables that consist of numbers as numeric. ] .pull-right[ ] .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M160 80c0-26.5 21.5-48 48-48h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H208c-26.5 0-48-21.5-48-48V80zM0 272c0-26.5 21.5-48 48-48H80c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V272zM368 96h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H368c-26.5 0-48-21.5-48-48V144c0-26.5 21.5-48 48-48z"/></svg> **An advanced stats mistake:** Modelling categorical, ordinal data as if it were numeric. ] .pull-right[ ] .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M192 32c17.7 0 32 14.3 32 32V199.5l111.5-66.9c15.2-9.1 34.8-4.2 43.9 11s4.2 34.8-11 43.9L254.2 256l114.3 68.6c15.2 9.1 20.1 28.7 11 43.9s-28.7 20.1-43.9 11L224 312.5V448c0 17.7-14.3 32-32 32s-32-14.3-32-32V312.5L48.5 379.4c-15.2 9.1-34.8 4.2-43.9-11s-4.2-34.8 11-43.9L129.8 256 15.5 187.4c-15.2-9.1-20.1-28.7-11-43.9s28.7-20.1 43.9-11L160 199.5V64c0-17.7 14.3-32 32-32z"/></svg> **A foundational stats mistake:** Interpreting a significant *p*-value as evidence that an effect exists. ] .pull-right[ ] --- count:false .pull-left[ ## The mistake ] .pull-right[ ## How you'll avoid it ] .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3L562.7 256l-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg> **A common R mistake:** Letting R treat all variables that consist of numbers as numeric. ] .pull-right[ When you know a variable is categorical, tell R that using `factor()`. ] .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M160 80c0-26.5 21.5-48 48-48h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H208c-26.5 0-48-21.5-48-48V80zM0 272c0-26.5 21.5-48 48-48H80c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V272zM368 96h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H368c-26.5 0-48-21.5-48-48V144c0-26.5 21.5-48 48-48z"/></svg> **An advanced stats mistake:** Modelling categorical, ordinal data as if it were numeric. ] .pull-right[ ] .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M192 32c17.7 0 32 14.3 32 32V199.5l111.5-66.9c15.2-9.1 34.8-4.2 43.9 11s4.2 34.8-11 43.9L254.2 256l114.3 68.6c15.2 9.1 20.1 28.7 11 43.9s-28.7 20.1-43.9 11L224 312.5V448c0 17.7-14.3 32-32 32s-32-14.3-32-32V312.5L48.5 379.4c-15.2 9.1-34.8 4.2-43.9-11s-4.2-34.8 11-43.9L129.8 256 15.5 187.4c-15.2-9.1-20.1-28.7-11-43.9s28.7-20.1 43.9-11L160 199.5V64c0-17.7 14.3-32 32-32z"/></svg> **A foundational stats mistake:** Interpreting a significant *p*-value as evidence that an effect exists. ] .pull-right[ ] --- ## Model ordinal data with `polr()` polr = **P**roportional **O**dds **L**ogistic **R**egression -- ``` r library(MASS) # MASS contains the polr() function anx_fit1 <- polr( rating ~ 1, # intercept-only model, to start data = anx, Hess = TRUE, method = 'probit' # ask me in the Q+A! ) ``` --- ## Model ordinal data with `polr()` ``` r summary(anx_fit1) ``` ``` ## Call: ## polr(formula = rating ~ 1, data = anx, Hess = TRUE, method = "probit") ## ## No coefficients ## ## Intercepts: ## Value Std. Error t value ## 1|2 -0.8420 0.0157 -53.7268 ## 2|3 -0.1678 0.0138 -12.1462 ## 3|4 0.3833 0.0141 27.1512 ## 4|5 1.0339 0.0168 61.6193 ## ## Residual Deviance: 26596.28 ## AIC: 26604.28 ``` --- ## What do those `Intercepts` mean? -- <img src="data:image/png;base64,#demo_files/figure-html/plot-underlying-normal-1.png" width="864" style="display: block; margin: auto;" /> ??? - imagine that there's some underlying continuous normal distribution of anxiety, assumed standard normal [show normal distrib] - ppl with high anxiety are more likely to give high responses, ppl with low anxiety more likely to give low responses (could do emojis relating to anxiety: <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 0 48 256a208 208 0 1 0 416 0zM0 256a256 256 0 1 1 512 0A256 256 0 1 1 0 256zm177.6 62.1C192.8 334.5 218.8 352 256 352s63.2-17.5 78.4-33.9c9-9.7 24.2-10.4 33.9-1.4s10.4 24.2 1.4 33.9c-22 23.8-60 49.4-113.6 49.4s-91.7-25.5-113.6-49.4c-9-9.7-8.4-24.9 1.4-33.9s24.9-8.4 33.9 1.4zm40-89.3l0 0 0 0-.2-.2c-.2-.2-.4-.5-.7-.9c-.6-.8-1.6-2-2.8-3.4c-2.5-2.8-6-6.6-10.2-10.3c-8.8-7.8-18.8-14-27.7-14s-18.9 6.2-27.7 14c-4.2 3.7-7.7 7.5-10.2 10.3c-1.2 1.4-2.2 2.6-2.8 3.4c-.3 .4-.6 .7-.7 .9l-.2 .2 0 0 0 0 0 0c-2.1 2.8-5.7 3.9-8.9 2.8s-5.5-4.1-5.5-7.6c0-17.9 6.7-35.6 16.6-48.8c9.8-13 23.9-23.2 39.4-23.2s29.6 10.2 39.4 23.2c9.9 13.2 16.6 30.9 16.6 48.8c0 3.4-2.2 6.5-5.5 7.6s-6.9 0-8.9-2.8l0 0 0 0zm160 0l0 0-.2-.2c-.2-.2-.4-.5-.7-.9c-.6-.8-1.6-2-2.8-3.4c-2.5-2.8-6-6.6-10.2-10.3c-8.8-7.8-18.8-14-27.7-14s-18.9 6.2-27.7 14c-4.2 3.7-7.7 7.5-10.2 10.3c-1.2 1.4-2.2 2.6-2.8 3.4c-.3 .4-.6 .7-.7 .9l-.2 .2 0 0 0 0 0 0c-2.1 2.8-5.7 3.9-8.9 2.8s-5.5-4.1-5.5-7.6c0-17.9 6.7-35.6 16.6-48.8c9.8-13 23.9-23.2 39.4-23.2s29.6 10.2 39.4 23.2c9.9 13.2 16.6 30.9 16.6 48.8c0 3.4-2.2 6.5-5.5 7.6s-6.9 0-8.9-2.8l0 0 0 0 0 0z"/></svg>, <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M256 48a208 208 0 1 0 0 416 208 208 0 1 0 0-416zM512 256A256 256 0 1 1 0 256a256 256 0 1 1 512 0zM168 320c-13.3 0-24 10.7-24 24s10.7 24 24 24h8V320h-8zm40 48h32V320H208v48zm96 0V320H272v48h32zm32 0h8c13.3 0 24-10.7 24-24s-10.7-24-24-24h-8v48zM168 288H344c30.9 0 56 25.1 56 56s-25.1 56-56 56H168c-30.9 0-56-25.1-56-56s25.1-56 56-56zm-23.6-80a32 32 0 1 1 64 0 32 32 0 1 1 -64 0zm192-32a32 32 0 1 1 0 64 32 32 0 1 1 0-64z"/></svg>) - so to estimate how different anxiety levels translate to different responses on the 1--5 scale, we draw thresholds on that distribution [add thresholds] - ppl with anxiety in this bin will respond with 1, in this bin with 2, etc. - and those thresholds, the cutpoints btwn ratings, are the intercepts. - [show intercept estimates, put thoes same numbers on the thresholds] - normal distribution assumption is from method = probit. other methods assume other underlying distributions, but the idea of thresholds is the same. --- count: false ## What do those `Intercepts` mean? <img src="data:image/png;base64,#demo_files/figure-html/plot-underlying-normal2-1.png" width="864" style="display: block; margin: auto;" /> --- count: false ## What do those `Intercepts` mean? <img src="data:image/png;base64,#demo_files/figure-html/plot-underlying-normal3-1.png" width="864" style="display: block; margin: auto;" /> --- #### How does a student's gender affect how they respond to "Going to ask my statistics teacher for individual help with material I am having difficulty understanding"? -- .pull-left[ <img src="data:image/png;base64,#demo_files/figure-html/plot-gender-bars-1.png" width="504" style="display: block; margin: auto;" /> ] -- .pull-right[ <img src="data:image/png;base64,#demo_files/figure-html/normals-stacked-onlyfem-1.png" width="504" style="display: block; margin: auto;" /> ] --- .center[ <img src="data:image/png;base64,#demo_files/figure-html/fem-normal-solo-1.png" width="936" style="display: block; margin: auto;" /> ] --- count: false <img src="data:image/png;base64,#demo_files/figure-html/fem-mal-normals-1.png" width="936" style="display: block; margin: auto;" /> --- count: false <img src="data:image/png;base64,#demo_files/figure-html/all-gender-normals-1.png" width="936" style="display: block; margin: auto;" /> --- .pull-left[ ## The mistake ] .pull-right[ ## How you'll avoid it ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3L562.7 256l-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg> **A common R mistake:** Letting R treat all variables that consist of numbers as numeric. ] .pull-right[ When you know a variable is categorical, tell R that using `factor()`. ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M160 80c0-26.5 21.5-48 48-48h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H208c-26.5 0-48-21.5-48-48V80zM0 272c0-26.5 21.5-48 48-48H80c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V272zM368 96h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H368c-26.5 0-48-21.5-48-48V144c0-26.5 21.5-48 48-48z"/></svg> **An advanced stats mistake:** Modelling categorical, ordinal data as if it were numeric. ] .pull-right[ ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M192 32c17.7 0 32 14.3 32 32V199.5l111.5-66.9c15.2-9.1 34.8-4.2 43.9 11s4.2 34.8-11 43.9L254.2 256l114.3 68.6c15.2 9.1 20.1 28.7 11 43.9s-28.7 20.1-43.9 11L224 312.5V448c0 17.7-14.3 32-32 32s-32-14.3-32-32V312.5L48.5 379.4c-15.2 9.1-34.8 4.2-43.9-11s-4.2-34.8 11-43.9L129.8 256 15.5 187.4c-15.2-9.1-20.1-28.7-11-43.9s28.7-20.1 43.9-11L160 199.5V64c0-17.7 14.3-32 32-32z"/></svg> **A foundational stats mistake:** Interpreting a significant *p*-value as evidence that an effect exists. ] .pull-right[ ] --- count: false .pull-left[ ## The mistake ] .pull-right[ ## How you'll avoid it ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3L562.7 256l-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg> **A common R mistake:** Letting R treat all variables that consist of numbers as numeric. ] .pull-right[ When you know a variable is categorical, tell R that using `factor()`. ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M160 80c0-26.5 21.5-48 48-48h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H208c-26.5 0-48-21.5-48-48V80zM0 272c0-26.5 21.5-48 48-48H80c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V272zM368 96h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H368c-26.5 0-48-21.5-48-48V144c0-26.5 21.5-48 48-48z"/></svg> **An advanced stats mistake:** Modelling categorical, ordinal data as if it were numeric. ] .pull-right[ Apply and interpret ordinal regression models (e.g., `polr()` from `MASS`). ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M192 32c17.7 0 32 14.3 32 32V199.5l111.5-66.9c15.2-9.1 34.8-4.2 43.9 11s4.2 34.8-11 43.9L254.2 256l114.3 68.6c15.2 9.1 20.1 28.7 11 43.9s-28.7 20.1-43.9 11L224 312.5V448c0 17.7-14.3 32-32 32s-32-14.3-32-32V312.5L48.5 379.4c-15.2 9.1-34.8 4.2-43.9-11s-4.2-34.8 11-43.9L129.8 256 15.5 187.4c-15.2-9.1-20.1-28.7-11-43.9s28.7-20.1 43.9-11L160 199.5V64c0-17.7 14.3-32 32-32z"/></svg> **A foundational stats mistake:** Interpreting a significant *p*-value as evidence that an effect exists. ] .pull-right[ ] --- ## Are the effects of `gender` significant? ``` ## Coefficients: ## Value Std. Error t value ## genderMale/Man -0.3280 0.03015 -10.880 ## genderAnother Gender 0.4846 0.11992 4.041 ``` No *p*-values in the model summary. -- But it's common practice to compare these *t*-values to a standard normal distribution, like *z*-scores. -- <img src="data:image/png;base64,#demo_files/figure-html/zscore-mm-1.png" width="648" style="display: block; margin: auto;" /> <img src="data:image/png;base64,#demo_files/figure-html/zscore-ag-1.png" width="648" style="display: block; margin: auto;" /> ??? Since both *p*-values are below 0.05: - we CAN reject the null hypothesis that gender has no effect on ratings. - **we CANNOT conclude that there really is an effect of gender.** --- ### Why don't significant *p*-values mean an effect exists? Because we can also get significant *p*-values when there really is *no* effect. -- .pull-left[ No difference in the true population: <img src="data:image/png;base64,#demo_files/figure-html/true-skew-probdist-1.png" width="504" style="display: block; margin: auto;" /> ] -- .pull-right[ A possible random sample (*n* = 50 per group): <img src="data:image/png;base64,#demo_files/figure-html/simdat-1.png" width="504" style="display: block; margin: auto;" /> ] --- ### Why don't significant *p*-values mean an effect exists? ``` r sim_fit <- polr(rating ~ group, data = simdat, method = 'probit', Hess = TRUE) summary(sim_fit) ``` ``` ## Coefficients: ## Value Std. Error t value ## groupGroup B -0.4479 0.2229 -2.009 ``` <br> -- <img src="data:image/png;base64,#demo_files/figure-html/zscore-sim-1.png" width="648" style="display: block; margin: auto;" /> <br> So *p* is significant, but in the true population, Group A and Group B were identical! --- .pull-left[ ## The mistake ] .pull-right[ ## How you'll avoid it ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3L562.7 256l-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg> **A common R mistake:** Letting R treat all variables that consist of numbers as numeric. ] .pull-right[ When you know a variable is categorical, tell R that using `factor()`. ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M160 80c0-26.5 21.5-48 48-48h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H208c-26.5 0-48-21.5-48-48V80zM0 272c0-26.5 21.5-48 48-48H80c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V272zM368 96h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H368c-26.5 0-48-21.5-48-48V144c0-26.5 21.5-48 48-48z"/></svg> **An advanced stats mistake:** Modelling categorical, ordinal data as if it were numeric. ] .pull-right[ Apply and interpret ordinal regression models (e.g., `polr()` from `MASS`). ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M192 32c17.7 0 32 14.3 32 32V199.5l111.5-66.9c15.2-9.1 34.8-4.2 43.9 11s4.2 34.8-11 43.9L254.2 256l114.3 68.6c15.2 9.1 20.1 28.7 11 43.9s-28.7 20.1-43.9 11L224 312.5V448c0 17.7-14.3 32-32 32s-32-14.3-32-32V312.5L48.5 379.4c-15.2 9.1-34.8 4.2-43.9-11s-4.2-34.8 11-43.9L129.8 256 15.5 187.4c-15.2-9.1-20.1-28.7-11-43.9s28.7-20.1 43.9-11L160 199.5V64c0-17.7 14.3-32 32-32z"/></svg> **A foundational stats mistake:** Interpreting a significant *p*-value as evidence that an effect exists. ] .pull-right[ ] --- count:false .pull-left[ ## The mistake ] .pull-right[ ## How you'll avoid it ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3L562.7 256l-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg> **A common R mistake:** Letting R treat all variables that consist of numbers as numeric. ] .pull-right[ When you know a variable is categorical, tell R that using `factor()`. ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M160 80c0-26.5 21.5-48 48-48h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H208c-26.5 0-48-21.5-48-48V80zM0 272c0-26.5 21.5-48 48-48H80c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V272zM368 96h32c26.5 0 48 21.5 48 48V432c0 26.5-21.5 48-48 48H368c-26.5 0-48-21.5-48-48V144c0-26.5 21.5-48 48-48z"/></svg> **An advanced stats mistake:** Modelling categorical, ordinal data as if it were numeric. ] .pull-right[ Apply and interpret ordinal regression models (e.g., `polr()` from `MASS`). ] <!-- --> .pull-left[ <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M192 32c17.7 0 32 14.3 32 32V199.5l111.5-66.9c15.2-9.1 34.8-4.2 43.9 11s4.2 34.8-11 43.9L254.2 256l114.3 68.6c15.2 9.1 20.1 28.7 11 43.9s-28.7 20.1-43.9 11L224 312.5V448c0 17.7-14.3 32-32 32s-32-14.3-32-32V312.5L48.5 379.4c-15.2 9.1-34.8 4.2-43.9-11s-4.2-34.8 11-43.9L129.8 256 15.5 187.4c-15.2-9.1-20.1-28.7-11-43.9s28.7-20.1 43.9-11L160 199.5V64c0-17.7 14.3-32 32-32z"/></svg> **A foundational stats mistake:** Interpreting a significant *p*-value as evidence that an effect exists. ] .pull-right[ Understand that significant *p*-values can arise even if no effect exists. ] -- <br> .center[**Thank you! <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:2em;width:2em;vertical-align:-0.5em;margin-left:0.2em;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 0 48 256a208 208 0 1 0 416 0zM0 256a256 256 0 1 1 512 0A256 256 0 1 1 0 256zm177.6 62.1C192.8 334.5 218.8 352 256 352s63.2-17.5 78.4-33.9c9-9.7 24.2-10.4 33.9-1.4s10.4 24.2 1.4 33.9c-22 23.8-60 49.4-113.6 49.4s-91.7-25.5-113.6-49.4c-9-9.7-8.4-24.9 1.4-33.9s24.9-8.4 33.9 1.4zm40-89.3l0 0 0 0-.2-.2c-.2-.2-.4-.5-.7-.9c-.6-.8-1.6-2-2.8-3.4c-2.5-2.8-6-6.6-10.2-10.3c-8.8-7.8-18.8-14-27.7-14s-18.9 6.2-27.7 14c-4.2 3.7-7.7 7.5-10.2 10.3c-1.2 1.4-2.2 2.6-2.8 3.4c-.3 .4-.6 .7-.7 .9l-.2 .2 0 0 0 0 0 0c-2.1 2.8-5.7 3.9-8.9 2.8s-5.5-4.1-5.5-7.6c0-17.9 6.7-35.6 16.6-48.8c9.8-13 23.9-23.2 39.4-23.2s29.6 10.2 39.4 23.2c9.9 13.2 16.6 30.9 16.6 48.8c0 3.4-2.2 6.5-5.5 7.6s-6.9 0-8.9-2.8l0 0 0 0zm160 0l0 0-.2-.2c-.2-.2-.4-.5-.7-.9c-.6-.8-1.6-2-2.8-3.4c-2.5-2.8-6-6.6-10.2-10.3c-8.8-7.8-18.8-14-27.7-14s-18.9 6.2-27.7 14c-4.2 3.7-7.7 7.5-10.2 10.3c-1.2 1.4-2.2 2.6-2.8 3.4c-.3 .4-.6 .7-.7 .9l-.2 .2 0 0 0 0 0 0c-2.1 2.8-5.7 3.9-8.9 2.8s-5.5-4.1-5.5-7.6c0-17.9 6.7-35.6 16.6-48.8c9.8-13 23.9-23.2 39.4-23.2s29.6 10.2 39.4 23.2c9.9 13.2 16.6 30.9 16.6 48.8c0 3.4-2.2 6.5-5.5 7.6s-6.9 0-8.9-2.8l0 0 0 0 0 0z"/></svg> Time for questions!**] --- count: false ## References Elazar, A., Alhama, R. G., Bogaerts, L., Siegelman, N., Baus, C., & Frost, R. (2022). When the "tabula" is anything but "rasa": What determines performance in the auditory statistical learning task? *Cognitive Science*, 46(2), e13102. Harrigan, K., Hogoboom, A., & Cochrane, L. (2022). Furthering student engagement: Lab sections in introductory linguistics. *Language*, 98(4), e199–e223. Reeder, P. A., Newport, E. L., & Aslin, R. N. (2017). Distributional learning of subcategories in an artificial grammar: Category generalization and subcategory restrictions. *Journal of Memory and Language*, 97, 17–29. Terry, J., Ross, R. M., Nagy, T., Salgado, M., Garrido-Vásquez, P., Sarfo, J. O., Cooper, S., Buttner, A. C., Lima, T. J. S., Öztürk, İ., Akay, N., Santos, F. H., Artemenko, C., Copping, L. T., Elsherif, M. M., Milovanović, I., Cribbie, R. A., Drushlyak, M. G., Swainston, K., … Field, A. P. (2023). Data from an International Multi-Centre Study of Statistics and Mathematics Anxieties and Related Variables in University Students (the SMARVUS Dataset). *Journal of Open Psychology Data*, 11(1), 8. --- count: false ## Helpful resources - Jamieson's (2004) paper _[Likert scales: How to (ab)use them](https://onlinelibrary.wiley.com/doi/10.1111/j.1365-2929.2004.02012.x)_ - UCLA Statistical Methods and Data Analytics's web page _[Ordinal Logistic Regression](https://stats.oarc.ucla.edu/r/dae/ordinal-logistic-regression/)_ - A. Solomon Kurz' (2021) blog post _[Notes on the Bayesian cumulative probit](https://stats.oarc.ucla.edu/r/dae/ordinal-logistic-regression/)_ - Gelman & Hill's (2007) book _[Data Analysis Using Regression and Multilevel/Hierarchical Models](https://www.cambridge.org/highereducation/books/data-analysis-using-regression-and-multilevel-hierarchical-models/32A29531C7FD730C3A68951A17C9D983)_ --- <!-- GRAVEYARD --> <!-- ## But first: --> <!-- Go to `menti.com` and enter code `3459 5977`. --> <!-- Or scan QR code: --> <!--  --> <!-- --- --> <!-- class: inverse, middle, center --> <!-- <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:5em;width:5em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M64 256V160H224v96H64zm0 64H224v96H64V320zm224 96V320H448v96H288zM448 256H288V160H448v96zM64 32C28.7 32 0 60.7 0 96V416c0 35.3 28.7 64 64 64H448c35.3 0 64-28.7 64-64V96c0-35.3-28.7-64-64-64H64z"/></svg> --> <!-- # The data we'll use --> <!-- --- --> <!-- ## The SMARVUS dataset (Terry et al., 2023) --> <!-- .center[SMARVUS = **S**tatistics and **M**athematics **A**nxieties and **R**elated **V**ariables in **U**niversity **S**tudents] --> <!-- -- --> <!-- .pull-left[ --> <!-- A survey of *n* = 18,841 students (mostly Psychology UGs) from 35 countries. --> <!-- Students rated their anxiety from 1 (no anxiety) to 5 (a great deal of anxiety) in scenarios like: --> <!-- - Studying for a statistics test. --> <!-- - Interpreting the meaning of a table in a journal article. --> <!-- - **Going to ask my statistics teacher for individual help with material I am having difficulty understanding. → ** --> <!-- ] --> <!-- -- --> <!-- .pull-right[ --> <!-- ```{r read-in-anx, include=F} --> <!-- anx <- read_csv('data/anx.csv') --> <!-- anx <- select(anx, unique_id, gender, score) |> --> <!-- rename(rating = score) |> --> <!-- mutate(gender = factor(gender, levels = c('Female/Woman', 'Male/Man', 'Another Gender'))) --> <!-- ``` --> <!-- ```{r bar-aggregated, echo=F, fig.width=7, fig.height=5.5} --> <!-- anx |> --> <!-- ggplot(aes(x = factor(rating))) + --> <!-- geom_bar(fill = '#2e3836') + --> <!-- theme_classic() + --> <!-- theme(text = element_text(family = "Fira Sans", size = 24)) + --> <!-- labs( --> <!-- x = element_blank(), --> <!-- y = 'Count', --> <!-- caption = 'n = 8,314' --> <!-- ) + --> <!-- scale_x_discrete(labels = c('1\n(no anxiety)', '2', '3', '4', '5\n(a great deal\nof anxiety)')) + --> <!-- NULL --> <!-- ``` --> <!-- ] --> <!-- --- --> <!-- class: inverse, middle, center --> <!-- <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:5em;width:5em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M416 208c0 45.9-14.9 88.3-40 122.7L502.6 457.4c12.5 12.5 12.5 32.8 0 45.3s-32.8 12.5-45.3 0L330.7 376c-34.4 25.2-76.8 40-122.7 40C93.1 416 0 322.9 0 208S93.1 0 208 0S416 93.1 416 208zm-312 8v64c0 13.3 10.7 24 24 24s24-10.7 24-24l0-64c0-13.3-10.7-24-24-24s-24 10.7-24 24zm80-96V280c0 13.3 10.7 24 24 24s24-10.7 24-24V120c0-13.3-10.7-24-24-24s-24 10.7-24 24zm80 64v96c0 13.3 10.7 24 24 24s24-10.7 24-24V184c0-13.3-10.7-24-24-24s-24 10.7-24 24z"/></svg> --> <!-- # Modelling an ordinal variable --> <!-- ### The .mono-white[polr()] express --> <!-- --- --> <!-- class: inverse, middle, center --> <!-- <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:5em;width:3.75em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M192 32c17.7 0 32 14.3 32 32V199.5l111.5-66.9c15.2-9.1 34.8-4.2 43.9 11s4.2 34.8-11 43.9L254.2 256l114.3 68.6c15.2 9.1 20.1 28.7 11 43.9s-28.7 20.1-43.9 11L224 312.5V448c0 17.7-14.3 32-32 32s-32-14.3-32-32V312.5L48.5 379.4c-15.2 9.1-34.8 4.2-43.9-11s-4.2-34.8 11-43.9L129.8 256 15.5 187.4c-15.2-9.1-20.1-28.7-11-43.9s28.7-20.1 43.9-11L160 199.5V64c0-17.7 14.3-32 32-32z"/></svg> --> <!-- # Interpreting *p*-values --> <!-- --- --> <!-- class: inverse, middle, center --> <!-- <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:5em;width:6.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M504 0c-9.7 0-18.5 5.8-22.2 14.8s-1.7 19.3 5.2 26.2l39 39-22.2 22.2C475.9 78.4 439.6 64 400 64c-88.4 0-160 71.6-160 160c0 80.2 59 146.6 136 158.2V408H352c-13.3 0-24 10.7-24 24s10.7 24 24 24h24v32c0 13.3 10.7 24 24 24s24-10.7 24-24V456h24c13.3 0 24-10.7 24-24s-10.7-24-24-24H424V382.2c77-11.6 136-78 136-158.2c0-31.4-9-60.7-24.7-85.4L560 113.9l39 39c6.9 6.9 17.2 8.9 26.2 5.2s14.8-12.5 14.8-22.2V24c0-13.3-10.7-24-24-24H504zM400 128a96 96 0 1 1 0 192 96 96 0 1 1 0-192zM190.9 18.1C188.4 12 182.6 8 176 8s-12.4 4-14.9 10.1l-29.4 74L55.6 68.9c-6.3-1.9-13.1 .2-17.2 5.3s-4.6 12.2-1.4 17.9l39.5 69.1L10.9 206.4c-5.4 3.7-8 10.3-6.5 16.7s6.7 11.2 13.1 12.2l78.7 12.2L90.6 327c-.5 6.5 3.1 12.7 9 15.5s12.9 1.8 17.8-2.6L176 286.1l58.6 53.9c4.1 3.8 9.9 5.1 15.2 3.6C223.6 310.8 208 269.2 208 224c0-60.8 28.3-115 72.4-150.2L220.3 92.1l-29.4-74z"/></svg> --> <!-- # Stats anxiety and gender --> <!-- <br> <br> --> <!-- <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M32 0C14.3 0 0 14.3 0 32S14.3 64 32 64V75c0 42.4 16.9 83.1 46.9 113.1L146.7 256 78.9 323.9C48.9 353.9 32 394.6 32 437v11c-17.7 0-32 14.3-32 32s14.3 32 32 32H64 320h32c17.7 0 32-14.3 32-32s-14.3-32-32-32V437c0-42.4-16.9-83.1-46.9-113.1L237.3 256l67.9-67.9c30-30 46.9-70.7 46.9-113.1V64c17.7 0 32-14.3 32-32s-14.3-32-32-32H320 64 32zM288 437v11H96V437c0-25.5 10.1-49.9 28.1-67.9L192 301.3l67.9 67.9c18 18 28.1 42.4 28.1 67.9z"/></svg> **First:** Think to yourself about the questions. --> <!-- <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M0 24C0 10.7 10.7 0 24 0H360c13.3 0 24 10.7 24 24s-10.7 24-24 24h-8V67c0 40.3-16 79-44.5 107.5L225.9 256l81.5 81.5C336 366 352 404.7 352 445v19h8c13.3 0 24 10.7 24 24s-10.7 24-24 24H24c-13.3 0-24-10.7-24-24s10.7-24 24-24h8V445c0-40.3 16-79 44.5-107.5L158.1 256 76.5 174.5C48 146 32 107.3 32 67V48H24C10.7 48 0 37.3 0 24zM110.5 371.5c-3.9 3.9-7.5 8.1-10.7 12.5H284.2c-3.2-4.4-6.8-8.6-10.7-12.5L192 289.9l-81.5 81.5zM284.2 128C297 110.4 304 89 304 67V48H80V67c0 22.1 7 43.4 19.8 61H284.2z"/></svg> **Then:** Ask your neighbour what they think. What's their reasoning? What's yours? --> <!-- <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:0.2em;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M32 0C14.3 0 0 14.3 0 32S14.3 64 32 64V75c0 42.4 16.9 83.1 46.9 113.1L146.7 256 78.9 323.9C48.9 353.9 32 394.6 32 437v11c-17.7 0-32 14.3-32 32s14.3 32 32 32H64 320h32c17.7 0 32-14.3 32-32s-14.3-32-32-32V437c0-42.4-16.9-83.1-46.9-113.1L237.3 256l67.9-67.9c30-30 46.9-70.7 46.9-113.1V64c17.7 0 32-14.3 32-32s-14.3-32-32-32H320 64 32zM96 75V64H288V75c0 25.5-10.1 49.9-28.1 67.9L192 210.7l-67.9-67.9C106.1 124.9 96 100.4 96 75z"/></svg> **Afterward:** we'll look at the model's estimates together and discuss. --> <!-- ```{r anx_fit2-display, eval = FALSE} --> <!-- anx_fit2 <- polr( --> <!-- rating ~ gender, --> <!-- data = anx, --> <!-- method = 'probit', --> <!-- Hess = TRUE --> <!-- ) --> <!-- summary(anx_fit2) --> <!-- ``` --> <!-- ``` --> <!-- ## Coefficients: --> <!-- ## Value Std. Error t value --> <!-- ## genderMale/Man -0.3280 0.03015 -10.880 --> <!-- ## genderAnother Gender 0.4846 0.11992 4.041 --> <!-- ## --> <!-- ## Intercepts: --> <!-- ## Value Std. Error t value --> <!-- ## 1|2 -0.9045 0.0169 -53.5402 --> <!-- ## 2|3 -0.2246 0.0150 -14.9847 --> <!-- ## 3|4 0.3318 0.0151 21.9158 --> <!-- ## 4|5 0.9889 0.0176 56.2958 --> <!-- ``` -->